Dynamics of Nonlocal Fisher concentration points: a nonlinear analysis of Turing patterns
Résumé
The so-called 'nonlocal Fisher' model takes into account an influence neighborhood for inhibition in classical Fisher ecological invasion. In this area, it has been introduced to represent front propagation with redistributed resources. More recently it has also been proposed as the simpler model exhibiting Turing instability and the biological interpretation refers to adaptive evolution. One aspect of the present paper is to propose a nonlinear analysis of these Turing patterns. More precisely, we introduce a rescaled equation in order to take into account rare mutations (small diffusion). We analyze in which circumstances such a model exhibits stable patterns, among them the Dirac concentrations (that are interpreted as morphs in adaptive dynamics) are remarkable. We use a change of variables, similar to the phase in WKB method, that describes more accurately the phenomenon and leads to a constrained Hamilton-Jacobi equation. It allows us to interpret several features of the patterns, as the weights of the Dirac concentration points, the asymmetry variable regulating their velocities and other relevant quantities.
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